Why not keep it local and go with a Mosin-Nagant if you don't have one already? With a few modifications they're good range rifles and the calibre is only slightly smaller than the Lee Enfield. Heck, if you go for the Polish variant then you get a 7.92mm cartridge, but I imagine that's probably harder to get ammunition for.
If you're looking for Lapua territory though, try Accuracy International AWM, no idea of the costs or availability (not really a hunter myself) but it's what our lads use with a .338 and I do believe it holds the current world record for long distance sniping.
http://en.wikipedia.org/wiki/Accuracy_International_AWM
It's being replaced with the AX338 so it might be coming down in price as the Army offloads their material.
I've had the fortune to be able to handle an AWM (well, L96A1 in our parlance) and found it to be quite nicely balanced with good optics. Although, I must admit I was more impressed by the SUSAT on the L85A2 and how clear it was at multiple ranges without manual adjustment.